# Tetrad formalism for a metric with signature $(+,-,-,-)$

For example, consider the Minkowski metric in 4D $$\text{d}s^2 = -\text{d}t^2 + \text{d}r^2 + r^2 \text{d}\theta^2 + r^2 \sin^2θ \text{d}\phi^2$$

For this, we construct tetrads as follows:

For outgoing vector $$l_\mu$$, we take $$\frac{\text{d}t + \text{d}r}{\sqrt{2}}$$ so that $$l_\mu$$ = $$\frac{1}{\sqrt{2}}$$ $$(1,1,0,0)$$ and for $$n_\mu$$, we take $$\frac{\text{d}t - \text{d}r}{\sqrt{2}}$$ so that $$n_\mu$$ =$$\frac{1}{\sqrt{2}}(1,-1,0,0).$$

For metric signature $$(+,-,-,-)$$ i.e if the metric is $$\text{d}s^2 = -\text{d}t^2 + \text{d}r^2 + r^2 \text{d}\theta^2 + r^2 \sin^2θ \text{d}\phi^2$$, what will be the tetrads?

• It'll be the same. You just change the metric signature for the flat metric. Sep 27, 2021 at 11:45
• @Prahar I used the same covariant tetrads and from that find corresponding contravariant tetrads using the metric. But it is not giving a correct answer. Forex. For (+,-,-,-) we have g_αβ = l_α n_β + n_α l_β - m_α¯m_β - m_β ¯m_α. So using the same tetrads I am not getting correct metric components using this equation for the metric.
– apk
Sep 27, 2021 at 13:18
• There must be a mistake. In the original $-+++$ signature, the correct formula is $g_{\mu\nu} = - l_\mu n_\nu - l_\nu n_\mu + m_\mu {\bar m}_\nu + m_\nu {\bar m}_\nu$ which is the same as $- g_{\mu\nu} = l_\mu n_\nu + l_\nu n_\mu - m_\mu {\bar m}_\nu - m_\nu {\bar m}_\nu$ and $-g_{\mu\nu}$ is the metric in the $+---$ signature. How can the results not match (unless your original tetrads are wrong)? Sep 27, 2021 at 13:21
• Yes, there was some mistake in my calculations. thanks, I got it. It is the same. Will this hold for any metric or only for the flat metric? For any metric, if we change the metric signature, can we use the same covariant tetrads and then find the corresponding contravariant tetrads?
– apk
Sep 27, 2021 at 16:52
• This is NP formalism. Given that a space-time (3+1) admits spin bundle, you can construct this orthonormal frame ($l,n,m,\bar{m}$) such that metric takes the form as you have stated. So it's pretty general for all metrices modelling some physical system
– KP99
Sep 28, 2021 at 3:55